A well studies model of active particles is the RTP model in 1D for particles trapped between two walls.
In the overdamped regime, the model has been frequently studies and is well understood
[1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}. For underdamped dynamics,
the system is governed by the Kramers equation resulting in two coupled differential equations:
\(&&\dot{p}_+ = -v\frac{\partial p_+}{\partial x}+ \frac{1}{\tau _r} \frac{\partial }{\partial v}\left[ \left(v - v_0\right) p_+ \right] + \frac{D}{\tau _r^2} \frac{\partial ^2 p_+}{\partial v^2} - \frac{1}{2} \frac{1}{\tau } (p_+ - p_-)\nonumber \\&&\dot{p}_- = -v\frac{\partial p_-}{\partial x}+ \frac{1}{\tau _r} \frac{\partial }{\partial v}\left[ \left(v + v_0\right) p_- \right] + \frac{D}{\tau _r^2} \frac{\partial ^2 p_-}{\partial v^2} + \frac{1}{2} \frac{1}{\tau } (p_+ - p_-),\nonumber \\\) 
